It is proved that a Musielak-Orlicz space L of real valued functions which is isometric to a Hilbert space coincides with L up to a weight, that is Φ(u,t) = c(t) u. Moreover it is shown that any surjective isometry between L and L is a weighted composition operator and a criterion for L to be isometric to L is presented.
We study order convexity and concavity of quasi-Banach Lorentz spaces , where 0 < p < ∞ and w is a locally integrable positive weight function. We show first that contains an order isomorphic copy of . We then present complete criteria for lattice convexity and concavity as well as for upper and lower estimates for . We conclude with a characterization of the type and cotype of in the case when is a normable space.
We show that under some assumptions on the Musielak−Orlicz function generating a quasi-Banach Musielak−Orlicz function space, the Banach envelope of the weighted Cesàro−Musielak−Orlicz space generated by a certain positive sublinear operator is a weighted -space.
We identify the class of Caldern-Lozanovskii spaces that do not contain an asymptotically isometric copy of , and consequently we obtain the corresponding characterizations in the classes of Orlicz-Lorentz and Orlicz spaces equipped with the Luxemburg norm. We also give a complete description of order continuous Orlicz-Lorentz spaces which contain (order) isometric copies of for each integer . As an application we provide necessary and sufficient conditions for order continuous Orlicz-Lorentz...
We investigate the relationships between strongly extreme, complex extreme, and complex locally uniformly rotund points of the unit ball of a symmetric function space or a symmetric sequence space E, and of the unit ball of the space E(ℳ,τ) of τ-measurable operators associated to a semifinite von Neumann algebra (ℳ,τ) or of the unit ball in the unitary matrix space . We prove that strongly extreme, complex extreme, and complex locally uniformly rotund points x of the unit ball of the symmetric...
For an Orlicz function φ and a decreasing weight w, two intrinsic exact descriptions are presented for the norm in the Köthe dual of the Orlicz-Lorentz function space or the sequence space , equipped with either the Luxemburg or Amemiya norms. The first description is via the modular , where f* is the decreasing rearrangement of f, ≺ denotes submajorization, and φ⁎ is the complementary function to φ. The second description is in terms of the modular ,where (f*)⁰ is Halperin’s level function...
We give sufficient and necessary conditions for complex extreme points of the unit ball of Orlicz-Lorentz spaces, as well as we find criteria for the complex rotundity and uniform complex rotundity of these spaces. As an application we show that the set of norm-attaining operators is dense in the space of bounded linear operators from into d(w,1), where is a predual of a complex Lorentz sequence space d(w,1), if and only if wi ∈ c₀∖ℓ₂.
This is a review article of geometric properties of noncommutative symmetric spaces of measurable operators , where is a semifinite von Neumann algebra with a faithful, normal, semifinite trace , and is a symmetric function space. If is a symmetric sequence space then the analogous properties in the unitary matrix ideals are also presented. In the preliminaries we provide basic definitions and concepts illustrated by some examples and occasional proofs. In particular we list and discuss...
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